Let be a first-order language and for each let be a class of -structures.
Show that .

Here are some basic definitions from the book "Model Theory", W. Hodges (which could help):

A sentence is a formula with no free variables. A theory is a set of sentences.
We say that is a model of , or that is true in , when holds. Given a theory in , we say that is a model of , in symbols , if is a model of every sentence in .
Let be a language and a class of -structures. We define the -theory of , , to be the set (or class) of all sentences of such that for every structure in . We omit the subscript when is first-order: the theory of , , is the set of all first-order sentences which are true in every structure in .

Thanks for any help.

November 17th 2010, 04:40 AM

emakarov

Quote:

Originally Posted by Arczi1984

Show that .

Consider two structures A and B. Then . Thus, both sides in the quote above consist of sentences that are true in all structures.

In more detail, show that for every , and that .

November 19th 2010, 08:22 AM

Arczi1984

I understand the first part. But how can I show, for example:

using this fact?

November 19th 2010, 09:21 AM

emakarov

Suppose . By definition of Th, is true in every structure of . In particular, is true in every structure of , i.e., .